The Einstein tensor in linearized perturbation GR is [1]}
\(G_{\mu \nu }= R_{\mu \nu } - \frac{1}{2}\eta _{\mu \nu }R= \frac{1}{2}( \partial _{\sigma } \partial _{\nu } h^{\sigma }_{\,\,\,\mu } + \partial _\sigma \partial _\mu h^{\sigma }_{\,\,\,\nu } - \partial _{\mu }\partial _{\nu }h -\Box h_{\mu \nu } -\eta _{\mu \nu }\partial _{\rho }\partial _{\lambda }h^{\rho \lambda } + \eta _{\mu \nu }\Box h )\,.\) 
